Problem: Factor the following expression: $9$ $x^2$ $-52$ $x+$ $35$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(9)}{(35)} &=& 315 \\ {a} + {b} &=& & & {-52} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $315$ and add them together. The factors that add up to ${-52}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-7}$ and ${b}$ is ${-45}$ $ \begin{eqnarray} {ab} &=& ({-7})({-45}) &=& 315 \\ {a} + {b} &=& {-7} + {-45} &=& -52 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {9}x^2 {-7}x {-45}x +{35} $ Group the terms so that there is a common factor in each group: $ ({9}x^2 {-7}x) + ({-45}x +{35}) $ Factor out the common factors: $ x(9x - 7) - 5(9x - 7) $ Notice how $(9x - 7)$ has become a common factor. Factor this out to find the answer. $(9x - 7)(x - 5)$